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Highway Trafﬁc Operations

-3

where

is measured in vehicles per mile per one lane,

is the length of the segment expressed in miles,

and

is the number of trafﬁc lanes. Typically, density is measured in each direction separately. The speeds

of vehicles at the moment of measurement can be averaged to calculate the so-called

space-mean speed S

(64.2)

where

= calculated based on

instantaneous observations

= the speed of vehicle

on the highway segment

Spot observations render the number of vehicles passing a spot during an observation period. This

count can be converted to

trafﬁc volume

(64.3)

where

is the volume typically expressed in vehicles per hour, estimated based on the count

observed

for period

(in hours). The speeds of vehicles are sometimes measured in spot observations, and the

average speed, called

time-mean speed

, is calculated as follows:

(64.4)

where

is the time-mean speed calculated based on

spot measurements and

is the speed of vehicle

included in the observation. The time-mean and space-mean speeds measured for the same trafﬁc are

typically not equal. This phenomenon will be explained later in this chapter.

Spot observations must be planned in accordance with the time of day, week, and year that are to be

represented by the measurements. Due to the considerable random and periodic variability of trafﬁc, the

observation period must be adequate for the purpose. Trafﬁc observations in short intervals may vary

randomly; thus a short observation period of seconds or minutes is not recommended to represent an

entire hour. On the other hand, trafﬁc ﬂuctuations over the period of an entire day make a several-hour

trafﬁc observation nonrepresentative for any particular hour within the observation period as well as for

the entire day. It is prudent to observe trafﬁc for 1 hour to obtain hourly trafﬁc estimates and for 1 day

for daily trafﬁc estimates. Of course, if the periodic ﬂuctuation is known, then even hourly observations

can be converted to daily observations by using appropriate conversion factors.

The Manual of Trafﬁc

Engineering Studies

(Robertson et al., 1994) and McShane et al. (1998) provide more information on

trafﬁc variability, adequate observation periods, and various trafﬁc studies.

Fundamental Trafﬁc Flow Relationship

Let us assume that all vehicles move at the same speed

. The time headway between two consecutive

vehicles

is the distance

between these two vehicles divided by speed

. If someone measures

the time headways and the distances between consecutive vehicles and calculates the average values

–

and

–

, the relationship between the average values will be preserved:

–

. It is also true that 1/

–

Please notice that the reverse of average intervehicle time is volume

and that the reverse of average

intervehicle distance is density

. Thus, we can write

(64.5)

Now, let us assume that a trafﬁc ﬂow consists of

classes of vehicles. All vehicles of class

move at

speed

. If

is the volume of

-class vehicles and

is the density of

-class vehicles, then we can claim

that

. Since the total volume is

VSD= ◊

-4

The Civil Engineering Handbook, Second Edition

and the total density is

we can say that

Since

is the space-mean speed

(densities are obtained in instantaneous observations), it is proven that

(64.6)

This relationship is called the fundamental trafﬁc ﬂow relationship.

Estimating Space-Mean Speed and Trafﬁc Density from Spot Observations

The source of discrepancy between the time-mean and space-mean speeds can be demonstrated by

considering a segment of highway with two classes of vehicles: slow and fast (Fig. 64.2 and Table 64.1).

The difference between the two speeds is caused by the higher fraction of fast vehicles in total volume

than in total density. Although the difference in the two speeds is negligible in typical trafﬁc conditions,

it may be considerable in a particular situation, such as when some vehicles are stopped. Zero speeds

signiﬁcantly reduce the average space-mean speed, but they are not included in the time-mean speed,

since the volume of the vehicles with zero speed is zero.

Spot observations render speeds of vehicles that pass a particular spot. Straight averaging gives the

time-mean speed. To estimate the space-mean speed, let us ﬁrst consider vehicles having the same speed.

Let

be a count of vehicles with speed

. The corresponding volume can be estimated as

= N

/T.

FIGURE 64.2 Example of a highway section for speeds comparison.

VSDDS

= ◊ =

ÂÂ

VSD

= ◊

0.25 mi

Slow vehicle, S

= 50 mi/h

Fast vehicle, S

= 70 mi/h

Highway Trafﬁc Operations 64-5

Using the relationship derived for ﬂows with the same speed, V

= S

, we can claim that D

= V

= N

/(S

·T). Since the space-mean speed is

we can say that

Since T and some speeds S

cancel out, the formula becomes

The result will not change if each vehicle constitutes its own class, even if some other vehicles have

the same speed. The ﬁnal formula is

(64.7)

The density near the spot can be estimated using the fundamental trafﬁc ﬂow equation, D = V/S

(N/T)/S

, or, after using Eq. (64.7),

TABLE 64.1 Comparison of the Space-Mean and Time-Mean Speeds

Va riable Equation Calculation

Space-Mean Speed

Density of slow vehicles D1 = N

/L 4/0.25 = 16 veh./mi.

Density of fast vehicles D2 = N

/L 6/0.25/2 = 24 veh./mi.

Total density D 16 + 24 = 40 veh./mi.

Fraction of slow vehicles on segment D

/D 8/20 = 0.40

Fraction of fast vehicles on segment D

/D 12/20 = 0.60

Space-mean speed on segment S

·D

/D + S

·D

/D 50(0.40) + 70(0.60) = 62.0 mi./hr

Time-Mean Speed

Vo lume of slow vehicles V

= S

·D

50 _ 16 = 800 veh./hr

Vo lume of fast vehicles V

= S

·D

70 _ 24 = 1680 veh./hr

Total volume V = V

+ V

800 +1680 = 2480 veh./hr

Fraction of slow vehicles in volume V

/V 800/2480 = 0.323

Fraction of fast vehicles in volume V

/V 0.677

Time-mean speed at spot on segment S

= S

·V

/V + S

·V

/V 50(0.323) + 70(0.677) = 63.5 mi./hr

Note: veh. = vehicle.

Lii

SN ST

NST

iii

◊

()

ÂÂ

64-6 The Civil Engineering Handbook, Second Edition

(64.8)

Please notice that spot observations may not include vehicles with zero speeds.

64.3 Measuring Techniques

Instantaneous Observations

An instantaneous observation requires an observer to be at a sufﬁcient elevation to see all vehicles on an

observed highway segment. A tall building located near the highway segment is the least expensive

solution. However, in many cases tall buildings are not available, so aerial photography is used. An

equipped aircraft takes two photographs in short succession and the vehicle speeds are estimated from

their shifts along the highway segment. Instantaneous observations are expensive and are not often used.

As has been shown, spot observations may be used to estimate space-mean speed and density.

Spot Observations

In manual techniques, human observers count and classify vehicles and sometimes measure their speeds.

This technique is accurate but expensive. Machine measurements are less expensive, and numerous tech-

niques and technologies are available. This section brieﬂy overviews several commonly used alternatives.

Vehicles are detected by devices called detectors, which utilize various physical phenomena such as

perturbation of electromagnetic or magnetic ﬁelds, changes of pressure in rubbers tubes, generation of

electrical ﬁeld in piezoelectric materials, detection of energy reﬂected or generated by a vehicle, and the

Doppler phenomenon caused by a vehicle. The types of energy used in vehicle detection include almost

all ranges of electromagnetic and acoustic waves. The most popular detectors are electromagnetic loops

installed in the pavement, and video detectors that use the visible range of electromagnetic waves are

becoming popular.

Regardless of the detection technology used, the most popular technique is based on a detection zone,

which is the spot on the pavement selected when setting a detector. Some detectors count only vehicles,

while others measure the time vehicles are present in the detection zone. Most detectors can use more

than one detection zone. The dimensions of detection zones vary from very small (microsensors) to large

enough to span across several lanes and over a long distance. The detection zone dimensions depend on

the type of detector and its purpose. If trafﬁc measurements are the purpose, the detection zone should

be small. A traditional detection zone is a 6-foot-long square or hexagon that covers a single trafﬁc lane.

Let us ﬁrst consider a detector that uses a single detection zone and is able to measure the time when

vehicles are present in the detection zone. Figure 64.3 presents two vehicles passing the detection zone,

represented by two trajectories: the front bumpers and the rear bumpers. The presented detector is

capable of returning counts and vehicle presence information. Typical detectors calculate so-called detec-

tor occupancy, which is the percent of time when the detection zone was occupied by vehicles. If an

observation period is 5 minutes, 20% detector occupancy means that for a total time of 1 minute during

the observation period one or more vehicles were present in the detection zone. Typically, detectors

return data in consecutive intervals. Volume V is calculated with Eq. (64.3). The following equations

calculate space-mean speed S

and density D from count N and detector occupancy B, obtained for an

interval of length T:

(64.9)

(64.10)

Nl l

◊◊ +

()

◊

100

◊ +

()

100

Highway Trafﬁc Operations 64-7

where l

stands for vehicle length and l

stands for detection zone length. These lengths should be in miles

and T should be in hours to obtain speed in miles/hour and density in vehicles/mile/lane. Detector occu-

pancy B is expressed in percent. Equation (64.9) can be derived by starting with individual vehicle speeds

= (l

+ l

)/t

, where l

is the length of vehicle i and t

is the detector occupancy time caused by vehicle i.

Equation (64.7), used together with the deﬁnition of detector occupancy, yields Eq. (64.9). Equation (64.10)

is obtained from Eq. (64.9) by using the fundamental ﬂow relationship shown in Eq. (64.6).

The main disadvantage of single detection zones is that the vehicle length is not measured and therefore

has to be assumed. Detectors with a single detection zone are used mainly to count vehicles and to

evaluate the congestion level through detector occupancy — a quantity directly related to trafﬁc density,

as seen in Eq. (64.10).

Detectors with two detection zones are much more reliable in measuring vehicle speeds. The known

distance between the leading edges of two detection zones divided by the travel time between the two

edges estimates the speed of a vehicle. The vehicle length is estimated from its speed and the time it

occupies the detection zone. This information is used to classify vehicles.

Cumulative Counts

A new technique based on cumulative counts at spots was proposed for freeway trafﬁc by Cassidy and

Windover (1995), using count detectors located along a freeway section at a frequency of approximately

1/2 mile. Detectors are required in all trafﬁc lanes, and vehicle counting starts as the same vehicle passes

the consecutive spots. Cumulative counts over an extended period can be used to estimate the number

of vehicles present between two spots (segment occupancy), travel times between two spots, vertical

queues, and delays (see Fig. 64.4).

64.4 Relationships between Volume, Speed, and Density

Trafﬁc ﬂow characteristics — volume, speed, and density — do not vary independently from one another.

The fundamental trafﬁc ﬂow equation ties these three characteristics together and allows for calculating

the quantity of one of them when the other two are known. The fundamental equation was derived from

the basic properties of trafﬁc ﬂows without any assumptions about driver behavior. An additional

equation is needed in order to calculate two unknown ﬂow characteristics from one known.

FIGURE 64.3 Measurements with a single detector.

off

Distance

Time

occ

Trajectory of

-vehicle’s

front

Trajectory of

-vehicle’s

rear

64-8 The Civil Engineering Handbook, Second Edition

Theories

Multiple theories furnish a second equation derived from assumptions about how drivers respond to

trafﬁc situations or by assuming analogies between trafﬁc and ﬂows (Gartner et al., 1995). For example,

Pipes (1953) postulates a linear relationship between the speed and the distance between vehicles. If one

assumes also that drivers are identical and consistent, that they do not pass other vehicles, and that

particular trafﬁc conditions exist long enough for drivers to adjust their speeds, the following speed–density

relationship can be derived:

(64.11)

The speed of vehicle ﬂow S depends on maximum volume V

, trafﬁc density D, and maximum density

, called the jam density. Equation (64.11) inherits the weakness of the crude assumptions about driver

behavior. The attempt to apply this equation to low-density conditions fails, since the speed prediction

is unrealistically high.

Among other theories, one that is considered reasonable for all trafﬁc conditions is called Greenshields’

equation (Greenshields, 1935). It assumes that drivers adjust their speed to the speed of preceding vehicles,

but unlike Eq. (64.11), this response weakens with the square distance between vehicles. The equation is

(64.12)

The theory says that vehicles move at their maximum speed, called free-ﬂow speed S

, when the

highway is empty (D = 0), and the speed drops linearly with an increase in density and reaches zero at

FIGURE 64.4 Measurements with cumulative counts.

Period with

reduced capacity

Time

Cumulative

Number of

Vehicles

Travel time

Delay

Segment

occupancy

Vertical

queue

Free-flow

travel time

Cumulative counts

at the first spot

Cumulative counts

at the second spot

= ◊ -

Highway Trafﬁc Operations 64-9

jam density D

. The relationship between volume and density is a parabolic curve obtained after using

Eq. (64.5) for a ﬂow of vehicles with identical speeds:

(64.13)

Figure 64.5 shows three relationships between volume, speed, and density derived from both the

Greenshields’ and fundamental trafﬁc ﬂow equations. These relationships indicate the existence of trafﬁc

conditions with maximum ﬂow C. This value is considered the capacity of the segment, with trafﬁc

characterized by the curves. Also, free-ﬂow and jam conditions are shown on these graphs.

Recent observations of freeway trafﬁc and simpliﬁcations in trafﬁc modeling have led to a piecewise

linear relationship between volume and density (Daganzo, 1992; Newell, 1993): V = min {S

·D, V

·(1 –

D/D

)}. Surprisingly, this relationship can be easily obtained from Eq. (64.11) by using free-ﬂow speed

as an upper bound for speed S and then applying the fundamental relationship in Eq. (64.5).

Greenshields’ equation and others are quite useful in understanding the effect of driver behavior on

trafﬁc conditions. One plausible result is that drivers tend to slow down when density increases; another

is that capacity is reached at a particular density value. If density grows beyond the capacity, the loss in

volume caused by the speed decrease exceeds the gain caused by the density increase, and the volume

drops. Capacity is determined by the spacing drivers maintain between vehicles. Drivers select spacing

perceived as safe for the speed that persists in the capacity conditions. The second half of the volume–den-

sity curve and the bottom half of the speed–volume curve describe trafﬁc forced to move slower by some

ﬂow constraint located downstream. Drivers slow down accordingly and reduce their distances between

vehicles. Jam density indicates the distance drivers prefer between vehicles when stopped in a queue. A

comprehensive overview of ﬂow relationships can be found in May (1990) and Gardner et al. (1995).

Field Observations

The presented trafﬁc ﬂow relationships were derived from assumptions that fall short of the true complexity

of trafﬁc. Drivers are not identical; they drive different vehicles, change lanes, and behave inconsistently.

FIGURE 64.5 Greenshields’ trafﬁc ﬂow relationships.

Speed

Density

Volume

Speed

Volume

VDS

= ◊◊-

64-10 The Civil Engineering Handbook, Second Edition

It should not be a surprise that the trends in ﬁeld observations do not fully follow theories. Figure 64.6

presents example results of spot observations (Banks, 1989). One striking departure from the Green-

shields’ model is the small speed reduction for quite a wide range of volumes, which is caused mainly

by the opportunity for passing on multilane highways. As long as passing is possible, drivers maintain

their preferred speeds, and the average ﬂow speeds do not indicate any effect of growing volume or

density. Speed declines noticeably when trafﬁc volume approaches capacity.

Other departures from the existing trafﬁc ﬂow theories are two values of capacity reported by Hall and

Agyemang-Duah (1991). It seems that trafﬁc moving smoothly for a while can reach quite high values,

which indicates that drivers accept short distances between vehicles. On the other hand, when trafﬁc

discharges from a standing queue, its volume does not reach the capacity observed for moving ﬂows.

As mentioned earlier, so-called forced trafﬁc is observed where the density of trafﬁc is larger than the

density value in capacity conditions. Dense trafﬁc is caused by a downstream obstruction, which is called

bottleneck, the effects of which are apparent when trafﬁc volumes without a bottleneck would be higher

than the bottleneck capacity. Bottlenecks are caused by a reduced number of lanes, activities that distract

drivers, or trafﬁc controls such as stop signs and trafﬁc signals.

Measurements representing capacity and near-capacity conditions are more scattered than those rep-

resenting other conditions. These measurement points, lying mostly inside the expected speed–volume

curve, represent transient conditions where trafﬁc is changing from free-ﬂowing to forced or from forced

to free-ﬂowing. Flows in transient conditions consist of drivers who rapidly change speeds to adjust to

new conditions. Because the ﬂow relationships discussed here represent conditions in which drivers have

already reached their preferred speeds and spacing, transient conditions are not well described with these

relationships.

The widely used Highway Capacity Manual (HCM) contains methods of predicting the capacity of

highway segments and bottlenecks and their impacts on trafﬁc conditions (Transportation Research

Board, 2000). The chapter on freeway segments uses speed–volume relationships generalized from ﬁeld

observations (Fig. 64.7). These curves apply to speciﬁc conditions, called ideal. The HCM provides a

means to convert the actual trafﬁc volume in actual conditions to its equivalent value under ideal

conditions. The speed–volume and fundamental relationships allow for the calculating of the speed and

density from known trafﬁc volume persisting on an analyzed freeway segment. It should be mentioned

that the bottom part of the curve for congested and forced trafﬁc is not shown, since the HCM focuses

on conditions that are acceptable to motorists. Further details on highway capacity are presented in

Section 64.6.

FIGURE 64.6 Example detector observations. (From Banks, J.H., Transp. Res. Rec., 1225, 53, 1989.)

Highway Trafﬁc Operations 64-11

64.5 Queues and Delays at Bottlenecks

Points on the earlier introduced ﬂow relationship curves represent trafﬁc conditions ranging from free-

ﬂow to jam. The points represent a stable situation when drivers’ speeds and distances have already been

adjusted and these conditions are called steady state. The HCM methods for freeways and other types

of road segments apply to steady-state trafﬁc that persists during the busiest 15 minutes of design or

rush hour. This is acceptable for uniform road sections without bottlenecks, since actual 15-minute trafﬁc

measurements indeed can be approximately steady. A single steady state cannot properly represent

highway operations where trafﬁc conditions change rapidly. Such changes are observed upstream of a

highway bottleneck, where the congested trafﬁc of vehicles in the queue meets the noncongested trafﬁc

of arriving vehicles.

Shock Waves

The boundary between congested and noncongested trafﬁc where vehicles slow down rapidly is called a

shock wave. A shock wave is also created by vehicles that accelerate rapidly after being released from a

queue, for example, during a green signal. A shock wave can travel forward or backward. The end of a

growing queue moves backward while the end of a dense column of vehicles behind a truck can move

forward.

Speed w of a shock wave depends on the characteristics of trafﬁc states that meet each other at the

shock wave location:

(64.14)

A truck blocking trafﬁc is the example presented in Fig. 64.8. The truck’s speed is lower than the ﬂow

speed in capacity conditions and passing is not possible. The straight sections on the volume–density

curve connect points represent distinct trafﬁc states. The slopes of these sections equal the corresponding

shock waves separating the two trafﬁc states (Eq. (64.14)). The solid lines on the time-space diagram are

the shock waves’ trajectories separating different trafﬁc states. The time-space areas bounded by the shock

waves’ trajectories are arriving trafﬁc, behind-truck trafﬁc, free-ﬂow trafﬁc in front of the truck, and

FIGURE 64.7 Speed–volume relationships for ideal conditions. (From Transportation Research Board, Highway

Capacity Manual, Special Report 209, National Research Council, Washington, D.C., 2000.)

64-12 The Civil Engineering Handbook, Second Edition

capacity conditions. The only backward shock wave is the shock wave of accelerations that separates the

capacity conditions from the behind-truck conditions. The capacity conditions appear after the truck

exits the road and when vehicles at the front of the dense column accelerate to reach the speed and the

spacing between vehicles desired for the capacity conditions.

When the truck’s speed is lower than the speed of unaffected vehicles and is higher than the capacity

speed, vehicles decelerate when joining the trafﬁc behind the truck and accelerate back to the original

speed after being released in the front of the column. The wave of accelerations is forward. Capacity state

does not occur since its appearance would require vehicles released from the front of the column to

decelerate to reach the capacity speed, which is against the tendency of drivers to travel at their desired

speeds if possible. More about the theory of shock waves can be found in Lighthill and Whitham (1955),

Richards (1955), Gartner et al. (1995), and in most textbooks on trafﬁc ﬂow theory.

A slow-moving truck can be viewed as a moving bottleneck, while a jam caused by immobile obstruc-

tions, such as trafﬁc signals or temporary road closures, is called a stationary bottleneck. Trafﬁc directly

upstream of an obstruction does not move and accumulates in a queue at jam density. When the

obstruction ends, the trafﬁc released in the front of the queue accelerates and reaches capacity state

because the zero speed of trafﬁc behind the obstruction is apparently lower than the capacity speed.

Figure 64.9 presents all the trafﬁc states and shock waves associated with a temporary obstruction.

The dimensions of the triangle with jam conditions can be calculated using Eq. (64.14) and other

equations. The triangle’s side parallel to the time axis is equal to the interruption time t

, while the slope of

the triangle’s side representing the wave of decelerations (boundary between unaffected and jam trafﬁc) is

FIGURE 64.8 Shock waves caused by a slow vehicle.

FIGURE 64.9 Shock waves caused by trafﬁc interruption.

Flow

Density

Distance

Time

Behind-truck

traffic

Capacity

traffic

Free-flow

traffic

Behind-truck

traffic

Unaffected

traffic

Free-flow

traffic

Capacity traffic

Unaffected

traffic

Flow

Density

Distance

Time

Unaffected

traffic

Capacity

traffic

Traffic

jam

Free-flow

traffic

Capacity

traffic

Unaffected

traffic

Traffic

jam

Free-flow

traffic

Trajectories

of vehicles

Trajectories

of shock

waves